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Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test , Leibniz's rule , or the Leibniz criterion .
In particular, there is a systematic methodology to solve the numerical coefficients {(a n,b n)} N n = 1 that yield a minimax approximation or bound for the closely related Q-function : Q ( x ) ≈ Q̃ ( x ) , Q ( x ) ≤ Q̃ ( x ) , or Q ( x ) ≥ Q̃ ( x ) for x ≥ 0 .
In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors , direct methods would deliver an exact solution (for example, solving a linear system of equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } by Gaussian elimination ).
For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields [34] + + + = + + + = + + + = (+ + +), which is times the original series, so it would have a sum of half of the natural logarithm of 2. By ...
Thus detrending does not solve the estimation problem. In order to still use the Box–Jenkins approach, one could difference the series and then estimate models such as ARIMA, given that many commonly used time series (e.g. in economics) appear to be stationary in first differences. Forecasts from such a model will still reflect cycles and ...
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control , [ 1 ] and can be formulated to construct solutions in a memory-efficient, factored form.